Flow Rate Abstract: The aim of this paper is to analyse the flow rate of the Nolichucky River in Tennessee. Various mathematical models are compared so as to find the most appropriate one and then estimate some specific features such as the maximum, the mean and the rate of change. The data used throughout this analysis originates from the Nolichucky River in Tennessee between 27 October 2002 and 2 November 2002. The time is measured in hours past midnight, starting at 00:00 on 27 October and the flow is measured in cubic feet per second (cfs). Fig.1 shows the original data. Fig.1 A scatter plot of the data illustrates the relationship between the points. The flow is represented on the y-axis and time on the x-axis. This is shown in fig.2 1 1 Graph Plotted in Macintosh Microsoft Excel 2004. 1 Time 0 6 12 18 24 30 36 42 48 54 60 66 72 Flow 440 450 480 570 680 800 980 1090 1520 1920 1670 1440 1380 Time 78 84 90 96 102 108 114 120 126 132 138 144 Flow 1300 1150 1060 970 900 850 800 780 740 710 680 660
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Flow Rate
Abstract: The aim of this paper is to analyse the flow rate of the Nolichucky River in Tennessee. Various mathematical models are compared so as to find the most appropriate one and then estimate some specific features such as the maximum, the mean and the rate of change.
The data used throughout this analysis originates from the Nolichucky River in Tennessee
between 27 October 2002 and 2 November 2002. The time is measured in hours past
midnight, starting at 00:00 on 27 October and the flow is measured in cubic feet per second
(cfs). Fig.1 shows the original data.
Fig.1
A scatter plot of the data illustrates the relationship between the points. The flow is
represented on the y-axis and time on the x-axis. This is shown in fig.21
It is important to know that a graph plotting flow rate over time is often called a Hydrograph.
1 Graph Plotted in Macintosh Microsoft Excel 2004.
The graph shows that the flow rate was less then 500 cfs2 when something caused it to
increase parabolically until reaching a maximum value 50 hours later. After this, the flow rate
decreased rapidly for about 20 hours and then steadily fell towards its initial value during the
next 60 hours.
Before finding the line of best fit, it is possible to estimate the rate of change of the data. This
is done using a numerical method and allows us to roughly illustrate the behaviour of the flow
rate. Considering the first two points of the graph:
€
(t = 0, flow = 440) and (t = 6, flow = 450) ,
the change
€
dx of the x values is
€
6 − 0 = 6 and the change
€
dy of the y values is
€
450 − 440 =10.
As a result, the rate of change of the data is
€
dy
dx=
10
6=1.66 . The procedure is repeated for all
points of data and the results are plotted in Fig.3. (Note that the time value is the average
between the two original points)
The rate of change of the flow rate is therefore positive and increasing during the first 52
hours and negative and decreasing for the rest of the time.
2 Cubic feet per second
2
Fig.3
-60
-40
-20
0
20
40
60
80
0 50 100 150
Time
dy/dx
In order to accurately process and analyse the data, it is necessary to find the lines of best fit,
or the mathematical relationship between the points on the graph. Best-fit lines are estimated
using technology in programs such as Microsoft Excel or the TI-84 GDC calculator.
A straightforward linear correlation would not be accurate because the data shows distinct
changes in patterns. Moreover, the graph does not resemble to any known polynomial
function except the quartic, which is still far from the real correlation as shows model 1
Model 1 shows the quartic best fit:
€
y = 5 ×10−5 x 4 − 0.0132x 3 + 0.807x 2 + 6.2022x + 346.64 .
Note that even if the line follows the trends to a certain extent, it is still very far from the original values. Moreover, the line increases again when time equals around 130 hours whereas the data shows a continuous decrease.
It seems that there is no single function that would show behaviour similar to the original
data. (i.e. increasing parabolically, reaching a maximum, decreasing parabolically and then
decreasing steadily.) It is therefore necessary to split the data into two parts in order to
produce more accurate best-fit lines.
3
Model 1
y = 5E-05x 4 - 0.0132x 3 + 0.8078x 2 + 6.2022x +
0
500
1000
1500
2000
2500
0 20 40 60 80 100 120 140 160
Time (hours)
Flow (cfs)Best-Fit line
Estimating from the graph, the maximum value takes place when time equals 54 hours hence
points occurring before 54 hours belong to the increasing trend and points occurring after 54
hours belong to the decreasing trend. The composite function representing the original data
will be in the form of
€
Flow =f1(t) for 0 ≤ t ≤ 54
f2(t) for 144 ≥ t ≥ 54
⎫ ⎬ ⎭
⎧ ⎨ ⎩
with one function effective for
times before 54 hours and another for times after 54 hours (note that for more accuracy the
point
€
t = 54 belongs to both functions.) This technique enables us to use different types of
functions as best fit lines.
The first hypothesis was that each trend followed an exponential pattern as shown in Model 2
It is clear that an exponential correlation is not the best model for the river flow. Indeed, the
model’s greatest rate of flow is significantly bellow the data’s maximum value (
€
t = 54, flow =1920) while its shape does not closely follow the pattern. The “increasing
trend” best-fit is too steep at first and does not reflect the flow’s initial steady behaviour.
Moreover, its y-intercept is below the data’s lowest point. The “decreasing trend” line is also
inadequate since it shows a continuous decrease throughout the graph whereas the original
data decreases rapidly at first and then slower after a few hours.
4
Model 2
y = 370.33e0.0279x y = 3163.6e -
0
500
1000
1500
2000
2500
0 20 40 60 80 100 120 140 160Time (hours)
Flow (cfs)
Increasing trend
Decreasing trend
In addition to the straightforward graphical analysis, it is possible to calculate the average
distance between the original data and the model. Let us consider the first point of the data,
which is situated on the y-axis. (
€
t = 0, flow = 440) If the model function is defined as
€
f (x) = 370.33e0.0279x , then its flow value is
€
f (0) = 370.33 cfs. The distance between the
model and the actual data is
€
f (x) − f (0) = 370.33− 440 = 70 when rounding off adequately.
Calculating the distance for all points is long and tedious however, computer software such as
Excel make the task much more efficient. Fig 4 shows the tables used to compute the average
distance for both increasing and decreasing patterns in the first model.
Fig.4
The average distance between the points and the model is 75 units for the increasing function
and 54 units for the decreasing function. The average distance for the entire model is 65 units.
5
Time (hours) Flow (cfs) F(t) F(t)-Flow Av. distance0 440 370 70 756 450 438 12
Hence the average flow rate from 00:00 on 28 October to 00:00 on 2 November is
€
1215.444 +1228.503
2≈1222 cfs
This exact rate also occurs twice, once when the quadratic and the power functions equals the
mean value:
€
1222 = 0.6271t 2 − 8.399t + 476.36
t = 41.82
t ≈ 42
€
1222 =147191t−1.095
t = 79.48
t ≈ 80
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Hence the average flow between those two dates occurs at 18:00 on 28 October and 08:00 on
29 October.
Using the information extracted from the mathematical model of the data, we can suggest a
possible weather patter that would account for the shape of the graph. Because the river’s
average flow rate between the 28th and the 2nd was significantly higher than the initial and
final rates, the total volume of water that flowed seems to be greater than usual. This could be
caused by rainfall in the area, which would then be collected in the river. Moreover, the model
shows a steep increase in flow rate that reaches its maximum value within a few hours. A
strong and sudden precipitation is therefore likely to have caused the shape of the graph.
When heavy rainfall occurs on land that cannot absorb and store the water because it is
urbanised, saturated or too compact, it flows on the surface and reaches the river within a
short time3. This will then lead to a temporary but rapid increase in flow rate and sometimes
floods. If the rain comes to an end, the surface runoff disappears and the flow of the river
returns to its normal level. This “falling limb” is always more gentle than the rising limb
because the water infiltrated in the ground starts reaching the river and continues contributing
to its flow. The graph of the original data is therefore an example of a “storm hydrograph”
and can be compared to the typical one in Fig.94
3 http://www.uwsp.edu/geo/faculty/ritter/geog101/textbook/hydrosphere/surface_water.html4 Ecole Polytechnique Federale de Lausanne: http://hydram.epfl.ch
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Fig.9 shows the sketch of a typical storm hydrograph. The increase in flow is the rising limb, the peak is the maximum and the decrease is the falling limb, or flood recession. Note that subsurface flow arises after the peak because of water infiltration and the normal flow rate is the baseflow.
Model 4 shows the graph and model of the original data.
The model appears to be close to reality confirming our hypothesis. Nevertheless, there are
still several limitations that will affect the results.
The first and most important limitation is the transition between the rising and falling limb.
The model shows it as an abrupt change in gradient meaning the flow rises rapidly and then
suddenly falls with not intermediate stage in between. Most models, however, contain a
somewhat flattened peak since the runoff water may flow through different path and takes
some time to disappear completely. One way to solve this problem is to divide the model’s
composite function further and use another quadratic polynomial for the 3 highest points. On
the other hand, transitions between the various new functions could be even more ambiguous
and inaccurate hence this is a possible limit of technology.
Secondly, the model does not take into account the slight change in direction of the rising
limb at the 8th point of the graph. This point could have a great importance in a complex and
technical analysis as it could show, for example, that subsurface flow takes less time than
expected to reach the river and therefore adds to the surface runoff. Taking measurements of
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Model 4y = 0.6271x2 - 8.399x + y = 147191x-1.095
0
500
1000
1500
2000
2500
0 20 40 60 80 100 120 140 160Time (hours)
Flow (cfs)
Increasing trend
Decreasing trend
the flow at smaller time intervals is a possible solution to this limitation and it would greatly
improve the accuracy and reliability of the model.
Finally, we may attempt to apply a similar model and reasoning to a different set of data in
order to verify the hypothesis and analysis made in the discussion. The new data originates
from the Nolichucky River in Tennessee between 1st May 2007 and 20th May 2007. The flow
rate values are the averages of each day and are shown below5.
Note that some parameters of the data are changed. Here, 24 hours pass between each
measurement. The rise and fall of the flow rate also takes place over a much longer period of
time. (20 days)
Applying the same model to this set of data produces the graph in Model 5.